- Partitions
- Incidence Matrix
- Tearing
- Solution by Tearing
- Summarising
- Equation set as a Graph
- Bubble Point Calculation Revisited

Consider:

We note that:

*f*_{1}and*f*_{2}can be rearranged and solved sequentially for*x*_{1}and*x*_{2}- Even when
*x*_{1}and*x*_{2}are known,*f*_{3}to*f*_{5}cannot be reordered to enable*x*_{3},*x*_{4}and*x*_{5}to be obtained sequentially. - However,
*x*_{6}only appears in*f*_{6}, so that after we have solved*f*_{3}to*f*_{5}, then*x*_{5}will be known, and we can solve*f*_{6}on its own for*x*_{6}.

We can call the subset (*f*_{1},*f*_{2}) the *head*
of the equation set, and*f*_{6} the *tail*. (In general,
of course, the
equations need not have been numbered and ordered so that these appeared
at the top and bottom of the list of equations!)

The group (*f*_{3},*f*_{4},*f*_{6}) is called a *partition*
of the set of equations
and represents a subset that must be solved *simultaneously*. If we
delete *x*_{1} and*x*_{2}, which will be known after solving*f*_{1}
and*f*_{2}, the
partition is:

There are three equations in three unknowns, so our `work factor' is 3

The partition is identifiable as the the group of equations with above diagonal elements.

These can be solved as follows:

Rearrange *f*_{3} to give *x*_{5} , in terms of *x*_{3} , i.e:

Similarly

And finally

This procedure is referred to as **tearing** the set of
equations, reducing them here to solution for a single unknown *x*_{3} , called the **tear variable**.

The incidence matrix:

has **one** variable with a supra-diagonal element;
this tells us that we can
reduce the equations to solving for a single variable.

The procedure can be represented:

Clearly **any one** of *x*_{3} , *x*_{4} or *x*_{5} could have been
chosen as the unknown.

into:

, the

Partition:

Tail:

One and only one edge may **leave** an equation node.
One and only one edge may **enter** a variable node.
Partition cycle shown bold.

Given a liquid mixture ofncomponents with mol fraction composition , at pressureP, determine the temperatureT, and the compositiony_{i}of the vapour in equilibrium with the liquid.

P^{*}_{i} - P^{*}_{i}(T) |
= | 0 | (1) |

= 0 | (2) | ||

y_{i} - k_{i} x_{i} |
= 0 | (3) | |

= 0 | (4) |

The range of

With the equations in the above sequence and the output set, say for a
binary, chosen to be:

The incidence matrix is:

The final, summation, equation does not involve

Next - Section 3.5.3: Example Questions

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